A055496 -id:A055496 - cp's OEIS frontend

A163469 a(n) = A055496(n+1) - 2*A055496(n).

1, 1, 1, 1, 3, 3, 3, 3, 3, 9, 15, 11, 11, 3, 3, 5, 9, 63, 3, 15, 5, 11, 15, 31, 1, 3, 21, 13, 9, 9, 31, 39, 15, 7, 25, 21, 35, 11, 9, 39, 53, 53, 81, 21, 5, 23, 9, 39, 17, 21, 19, 3, 77, 5, 39, 57, 41, 29, 45, 21, 11, 15, 13, 39, 17, 33, 67, 129, 33, 13, 3, 5, 105, 67, 7, 13, 3, 15, 63

Offset: 1

Zak Seidov, Jul 28 2009

nonn

Michel Marcus, Table of n, a(n) for n = 1..999

Cf. A055496.

A006992 Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 5003, 9973, 19937, 39869, 79699, 159389, 318751, 637499, 1274989, 2549951, 5099893, 10199767, 20399531, 40799041, 81598067, 163196129, 326392249, 652784471, 1305568919, 2611137817

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

a(n) < a(n+1) by Bertrand's postulate (Chebyshev's theorem). - Jonathan Sondow, May 31 2014
Let b(n) = 2^n - a(n). Then b(n) >= 2^(n-1) - 1 and b(n) is a B_2 sequence: 0, 1, 3, 9, 19, 41, 85, 173, 349, ... - Thomas Ordowski, Sep 23 2014 See the link for B_2 sequence.
These primes can be obtained of exclusive form using a restricted variant of Rowland's prime-generating recurrence (A106108), making gcd(n, a(n-1)) = -1 when GCDs are greater than 1 and less than n (see program). These GCDs are also a divisor of each odd number from a(n) + 2 to 2*a(n-1) - 1 in reverse order, so that this subtraction with -1's invariably leads to the prime. - Manuel Valdivia, Jan 13 2015
First row of array in A229607. - Robert Israel, Mar 31 2015
Named after the French mathematician Joseph Bertrand (1822-1900). - Amiram Eldar, Jun 10 2021

Martin Aigner and Günter M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 7.
Martin Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), page 115. [From Martin Griffiths, Mar 28 2009]
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 344.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 189.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 100 terms from T. D. Noe)
Paul Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged, Vol. 5 (1932), pp. 194-198.
Paul Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., Vol. 9 (1934), pp. 282-288.
Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., Vol. 11 (1919), pp. 181-182.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq., Vol. 15 (2012) Article 12.5.4.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, Vol. 116, No. 7 (2009), pp. 630-635.
Jonathan Sondow and Eric Weisstein, MathWorld: Bertrand's Postulate.
Eric Weisstein's World of Mathematics, B2 Sequence.
Robert G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993.

Cf. A055496, A163961.
See A185231 for another version.
Cf. A007917, A229607, A295262.

a006992 n = a006992_list !! (n-1)
a006992_list = iterate (a007917 . (* 2)) 2
-- Reinhard Zumkeller, Sep 17 2014

A006992 := proc(n) option remember; if n=1 then 2 else prevprime(2*A006992(n-1)); fi; end;

bertrandPrime[1] = 2; bertrandPrime[n_] := NextPrime[ 2*a[n - 1], -1]; Table[bertrandPrime[n], {n, 40}]
(* Second program: *)
NestList[NextPrime[2#, -1] &, 2, 40] (* Harvey P. Dale, May 21 2012 *)
k = 3; a[n_] := If[GCD[n,k] > 1 && GCD[n, k] < n, -1, GCD[n, k]]; Select[Differences@Table[k = a[n] + k, {n, 2611137817}], # > 1 &] (* Manuel Valdivia, Jan 13 2015 *)

print1(t=2);for(i=2,60,print1(", "t=precprime(2*t))) \\ Charles R Greathouse IV, Apr 01 2013

from sympy import prevprime
l = [2]
for i in range(1, 51):
    l.append(prevprime(2 * l[i - 1]))
print(l) # Indranil Ghosh, Apr 26 2017

a(n+1) = A007917(2*a(n)). - Reinhard Zumkeller, Sep 17 2014

Limit_{n -> infinity} a(n)/2^n = 0.303976447924... - Thomas Ordowski, Apr 05 2015

Definition completed by Jonathan Sondow, May 31 2014

B_2 sequence link added by Wolfdieter Lang, Oct 09 2014

A051254 Mills primes.

Original entry on oeis.org

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499

Offset: 1

Simon Plouffe

Mills showed that there is a number A > 1 but not an integer, such that floor( A^(3^n) ) is a prime for all n = 1, 2, 3, ... A is approximately 1.306377883863... (see A051021).
a(1) = 2 and (for n > 1) a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006, corrected by M. F. Hasler, Sep 11 2024
The name refers to the American mathematician William Harold Mills (1921-2007). - Amiram Eldar, Jun 23 2021

			a(3) = 1361 = 11^3 + 30 = a(2)^3 + 30 and there is no smaller k such that a(2)^3 + k is prime. - _Jonathan Vos Post_, May 05 2006

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 130.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 137.

Robert G. Wilson v, Table of n, a(n) for n = 1..8
Chris K. Caldwell, Mills' Theorem - a generalization.
Chris K. Caldwell and Yuanyou Chen, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Steven R. Finch, Mills' Constant. [Broken link]
Steven R. Finch, Mills' Constant. [From the Wayback machine]
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, Amer. Math. Monthly, Vol. 126, No. 1 (2019), pp. 72-73; ResearchGate link, arXiv preprint, arXiv:2010.15882 [math.NT], 2020.
James Grime and Brady Haran, Awesome Prime Number Constant, Numberphile video, 2013.
Brian Hayes, Pumping the Primes, bit-player, Aug 19 2015.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
William H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53, No. 6 (1947), p. 604; Errata, ibid., Vol. 53, No 12 (1947), p. 1196.
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.
László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.
Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Mills' Prime.
Eric Weisstein's World of Mathematics, Prime Formulas.
Eric W. Weisstein, Table of n, a(n) for n = 1..13.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913.
Cf. A224845 (integer lengths of Mills primes).
Cf. A108739 (sequence of offsets b_n associated with Mills primes).
Cf. A051021 (decimal expansion of Mills constant).

floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. (A051021).

p = 1; Table[p = NextPrime[p^3], {6}] (* T. D. Noe, Sep 24 2008 *)
NestList[NextPrime[#^3] &, 2, 5] (* Harvey P. Dale, Feb 28 2012 *)

a(n)=if(n==1, 2, nextprime(a(n-1)^3)) \\ Charles R Greathouse IV, Jun 23 2017

apply( {A051254(n, p=2)=while(n--, p=nextprime(p^3));p}, [1..6]) \\ M. F. Hasler, Sep 11 2024

a(1) = 2; a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006

Edited by N. J. A. Sloane, May 05 2007

A059788 a(n) = largest prime < 2*prime(n).

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 37, 43, 53, 61, 73, 79, 83, 89, 103, 113, 113, 131, 139, 139, 157, 163, 173, 193, 199, 199, 211, 211, 223, 251, 257, 271, 277, 293, 293, 313, 317, 331, 337, 353, 359, 379, 383, 389, 397, 421, 443, 449, 457, 463, 467, 479, 499, 509, 523

Offset: 1

Labos Elemer, Feb 22 2001

nonn

Also, smallest member of the first pair of consecutive primes such that between them is a composite number divisible by the n-th prime. - Amarnath Murthy, Sep 25 2002

Except for its initial term, A006992 is a subsequence based on iteration of n -> A151799(2n). The range of this sequence is a subset of A065091. - M. F. Hasler, May 08 2016

			n=18: p(18)=61, so a(18) is the largest prime below 2*61=122, which is 113.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A005382, A005383, A005384, A005385, A055496, A006992, A052248.

with(numtheory):
A059788 := proc(n)
    prevprime(2*ithprime(n)) ;
end proc:
seq(A059788(n),n=1..50) ; # R. J. Mathar, May 08 2016

a[n_] := Prime[PrimePi[2Prime[n]]]
NextPrime[2*Prime[Range[100]], -1] (* Zak Seidov, May 08 2016 *)

a(n) = precprime(2*prime(n)); \\ Michel Marcus, May 08 2016

a(n) = A007917(A100484(n)). - R. J. Mathar, May 08 2016

A116533 a(1)=1, a(2)=2, for n > 2 if a(n-1) is prime, then a(n) = 2*a(n-1), otherwise a(n) = a(n-1) - 1.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 9, 8, 7, 14, 13, 26, 25, 24, 23, 46, 45, 44, 43, 86, 85, 84, 83, 166, 165, 164, 163, 326, 325, 324, 323, 322, 321, 320, 319, 318, 317, 634, 633, 632, 631, 1262, 1261, 1260, 1259, 2518, 2517, 2516, 2515, 2514, 2513, 2512, 2511, 2510, 2509, 2508

Offset: 1

Rodolfo Kurchan, Mar 26 2006

nonn

For n >= 3, using Wilson's theorem, a(n) = a(n-1) + (-1)^r*gcd(a(n-1), W), where W = A038507(a(n-1) - 1), and r=1 if gcd(a(n-1), W) = 1 and r=0 otherwise. - Vladimir Shevelev, Aug 07 2009

Cf. A006992, A055496, A080359, A104272, A106108, A132199. - Vladimir Shevelev, Aug 07 2009

a[1]:=1: a[2]:=2: for n from 3 to 60 do if isprime(a[n-1])=true then a[n]:=2*a[n-1] else a[n]:=a[n-1]-1 fi od: seq(a[n],n=1..60); # Emeric Deutsch, Apr 02 2006

More terms from Emeric Deutsch, Apr 02 2006

A163961 First differences of A116533.

Original entry on oeis.org

1, 2, -1, 3, -1, 5, -1, -1, -1, 7, -1, 13, -1, -1, -1, 23, -1, -1, -1, 43, -1, -1, -1, 83, -1, -1, -1, 163, -1, -1, -1, -1, -1, -1, -1, -1, -1, 317, -1, -1, -1, 631, -1, -1, -1, 1259, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 2503, -1, -1, -1, 5003, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Vladimir Shevelev, Aug 07 2009, Aug 14 2009

sign

Ignoring the +-1 terms, we obtain the sequence of Bertrand's primes A006992. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A116533, but with initials a_1(1)=2, a_2(1)=11, a_3(1)=17,..., a_m(1)=A164368(m),..., then the union of A_1,A_2,... contains all primes.

Cf. A116533, A006992, A055496, A080359, A104272, A106108, A132199, A164368

A116533 := proc(n) option remember; if n <=2 then n; else if isprime(procname(n-1)) then 2*procname(n-1) ; else procname(n-1)-1 ; end if; end if; end proc:
A163961 := proc(n) A116533(n+1)-A116533(n) ; end proc: # R. J. Mathar, Sep 03 2011

Differences@ Prepend[NestList[If[PrimeQ@ #, 2 #, # - 1] &, 2, 90], 1] (* Michael De Vlieger, Dec 06 2018 *)

a116533(n) = if(n==1, 1, if(n==2, 2, if(ispseudoprime(a116533(n-1)), 2*a116533(n-1), a116533(n-1)-1)))
a(n) = a116533(n+1)-a116533(n) \\ Felix Fröhlich, Dec 06 2018

lista(nn) = {va = vector(nn); va[1] = 1; va[2] = 2; for (n=3, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]-1);); vector(nn-1, n, va[n+1] - va[n]);} \\ Michel Marcus, Dec 07 2018

A163963 First differences of A080735.

Original entry on oeis.org

1, 2, 1, 5, 1, 11, 1, 23, 1, 47, 1, 1, 1, 97, 1, 1, 1, 197, 1, 1, 1, 397, 1, 1, 1, 797, 1, 1, 1, 1597, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3203, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12853, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25717, 1, 1, 1, 51437, 1, 1, 1

Offset: 1

Vladimir Shevelev, Aug 07 2009

nonn

Ignoring the 1 terms we obtain A055496. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A080735, but with initials a_1(1)=2, a_2(1)=3, a_3(1)=13,..., a_m(1)=A080359(m),..., then the union of A_1,A_2,... contains all primes.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A116533, A163961, A080735, A006992, A055496, A080359, A104272, A106108, A132199.

A080735 := proc(n) option remember; local p ; if n = 1 then 1; else p := procname(n-1) ; if isprime(p) then 2*p; else p+1 ; end if; end if; end proc: A163963 := proc(n) A080735(n+1)-A080735(n) ; end: seq(A163963(n),n=1..100) ; # R. J. Mathar, Nov 05 2009

Differences@ NestList[If[PrimeQ@ #, 2 #, # + 1] &, 1, 87] (* Michael De Vlieger, Dec 06 2018, after Harvey P. Dale at A080735 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]+1);); vector(nn-1, n, va[n+1] - va[n]);} \\ Michel Marcus, Dec 06 2018

More terms from R. J. Mathar, Nov 05 2009

A059786 Smallest prime after 2*(n-th prime).

Original entry on oeis.org

5, 7, 11, 17, 23, 29, 37, 41, 47, 59, 67, 79, 83, 89, 97, 107, 127, 127, 137, 149, 149, 163, 167, 179, 197, 211, 211, 223, 223, 227, 257, 263, 277, 281, 307, 307, 317, 331, 337, 347, 359, 367, 383, 389, 397, 401, 431, 449, 457, 461, 467, 479, 487, 503, 521

Offset: 1

Labos Elemer, Feb 22 2001

nonn

			n=17, 18, p(17)=59, p(18)=61, after 118 and 122 the next prime is 127, so a(17)=a(18)=127.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A055496, A006992.

with(numtheory): [seq(nextprime(2*ithprime(k)),k=1..256)];

NextPrime/@(2*Prime[Range[60]]) (* Harvey P. Dale, May 03 2019 *)

a(n) = nextprime(2*prime(n)+1); \\ Michel Marcus, Sep 21 2017

a(n) = A117928(n,1) for n>1. - Reinhard Zumkeller, Apr 03 2006

A076656 a(1) = 2; a(n) is smallest prime > 3*a(n-1).

Original entry on oeis.org

2, 7, 23, 71, 223, 673, 2027, 6089, 18269, 54829, 164503, 493523, 1480571, 4441721, 13325171, 39975553, 119926691, 359780077, 1079340313, 3238020943, 9714062893, 29142188683, 87426566057, 262279698173, 786839094529

Offset: 1

Cino Hilliard, Oct 24 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055496.

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; p=2;lst={p};Do[p=PrimeNext[3*p];AppendTo[lst,p],{n,2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, May 27 2009 *)
NestList[NextPrime[3#]&,2,30] (* Harvey P. Dale, Aug 12 2016 *)

Edited by Don Reble, May 03 2006

A065122 a(1) = 2; a(n) is smallest prime > 10*a(n-1).

Original entry on oeis.org

2, 23, 233, 2333, 23333, 233341, 2333459, 23334601, 233346041, 2333460413, 23334604169, 233346041759, 2333460417637, 23334604176379, 233346041763823, 2333460417638239, 23334604176382489, 233346041763824911

Offset: 1

Henry Bottomley, Nov 13 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A024770, A055496.

NextPrim[n_Integer] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; a[1] = 2; a[n_] := NextPrim[ 10*a[n - 1]]; Table[ a[n], {n, 1, 20} ]
NestList[NextPrime[10#]&,2,20] (* Harvey P. Dale, Jun 03 2012 *)

{ for (n=1, 100, if (n>1, a=nextprime(10*a), a=2); write("b065122.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 10 2009

More terms from Robert G. Wilson v, Nov 28 2001

cp's OEIS Frontend

A163469 a(n) = A055496(n+1) - 2*A055496(n).

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A006992 Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.

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A051254 Mills primes.

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A059788 a(n) = largest prime < 2*prime(n).

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A116533 a(1)=1, a(2)=2, for n > 2 if a(n-1) is prime, then a(n) = 2*a(n-1), otherwise a(n) = a(n-1) - 1.

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A163961 First differences of A116533.

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A163963 First differences of A080735.

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